Linear Transformations: Functions That Respect Geometry

A matrix becomes far easier to understand when we stop staring at its entries and ask:

What does this matrix do to space?

A linear transformation $T$ respects addition and scaling:

$$T(u+v)=T(u)+T(v),\qquad T(\alpha u)=\alpha T(u).$$

The columns tell the whole story

For

$$A= \begin{bmatrix} a&b\\ c&d \end{bmatrix},$$

the first column is $Ae_1$ and the second is $Ae_2$.

Because every vector is a combination of $e_1,e_2$, linearity determines what happens everywhere else.

Composition becomes multiplication

If $A$ acts first and $B$ second,

$$x\mapsto Ax\mapsto B(Ax)=(BA)x.$$

Matrix multiplication is therefore composition of geometric machines.

That interpretation makes the order $BA$ feel natural rather than arbitrary.